2 Answers
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You'll often hear that it's because of friction, but that's often not the main factor. For larger objects it's more likely the pressure they create.

In both cases the reason is the enormous speed, often tens of kilometers per second. When a larger object enters the atmosphere at these speed the air in front of it gets compressed so much that it becomes extremely hot. (Think of pumping up a tire; you're also compressing air and you can feel the valve becoming hot.) The compressed air will often disintegrate the object in the air, and then the debris may burn because of the heat. This is exactly what happened to the asteroid above Russia last year: it exploded with an enormous flash in the air, and left little traces on the ground.

This happens on other planets as well, if they have a sufficiently dense atmosphere. In 1994 the comet Shoemaker-Levy crashed into Jupiter. It disintegrated before entering Jupiter's atmosphere due to the strong gravitation, but when the fragments entered the atmosphere they could easily be seen lighting up as they burned up.

edit
Remember the Space Shuttle? It had heat resistant tiles on the bottom of the craft to protect it from burning when it entered the atmosphere, even though its speed is only a fraction of a meteorite's speed when that enters the atmosphere.
During the last launch of the Space Shuttle Columbia some material from the external fuel tank damaged this heat shield, and upon re-entry the heat and the highly pressurized air under the craft could enter it, causing the craft to disintegrate and kill all crew.

So if the speed and the air's pressure are the reason, such a thing won't happen in space itself where there is a vacuum right?
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A.KFeb 17 '14 at 13:58

@A.K - Right. Comets move at speeds of hundreds of kilometers per second without having any medium in which they would heat up. Note that comets may evaporate very slowly due to the sun's radiation though, but this is nothing mechanical.
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stevenvhFeb 17 '14 at 14:33

Good answer, any chance of getting some sources in there too?
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RhysW♦Feb 18 '14 at 8:37

@RhysW - 'fraid not. This is knowledge I collected over several decades through several sources, including courses. I read in meta that there seems to be a need for references for some people, but I disagree (as I pointed out there).
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stevenvhFeb 18 '14 at 11:10

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@stevenvh it doesn't necessarily have to be 'this is exactly where I got my knowledge from' as much as its good to have a source to say 'here is verification that this isn't just speculation on my behalf'. For example, is there any source somewhere that agrees with you on it being based more on pressure than friction? ect, hope this helps!
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RhysW♦Feb 18 '14 at 11:11

An object hitting the Earth at high speed has a lot of kinetic energy, \begin{align}
\frac{1}{2}mv^2
\end{align}
where m is the object's mass and v is it's velocity. If the velocity goes from lots to nothing relative to the Earth, that kinetic energy has to go somewhere. That somewhere ends up as heat.